I want to evaluate the Hurwitz zeta function
Φ(z,−4s,0)=∑k=0zkk−4s
And |z|<1 and s>1.
I want to have un upper bound for it.
I tried even Wolfram Mathematica (to have some hint of the form if possible fot the calculus) , but without success (since I give parameters and no numbers as an input).
Answer
The OP is asking about the upper bound of the function:
Φ(z,−4s,0)=∞∑k=1k4szk
|z|<1s>1
First, it's obvious that:
∞∑k=1k4szk≤∞∑k=1k4s|z|k
So let us consider only z>0.
For a fixed z we can see that:
p>q∞∑k=1k4pzk>∞∑k=1k4qzk
Which means that if an upper bound exists for Φ as a function of s, it will be the limit:
lim
Now let us fix a finite s and see what happens for z \to 1. For z=1 the series obviously diverges, which automatically means that for z close to 1 the value can be as large as we want, which means there's no upper bound for a fixed s either.
More rigorously, we need to prove that for any N>0 there exists \epsilon >0 such that:
\sum_{k=1}^\infty k^{4s} (1-\epsilon)^k > N
It's rather easy, we can just compare to geometric series:
\sum_{k=1}^\infty k^{4s} (1-\epsilon)^k>\sum_{k=1}^\infty (1-\epsilon)^k=\frac{1-\epsilon}{\epsilon}=\frac{1}{\epsilon}-1
Now pick \epsilon=\frac{1}{N+1} and the proof is finished.
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